Polyfit: Polynomial is badly conditioned

I have the following:

N = 5;

 for i=1:N
 p(i,:) = polyfit(time(:,3),values(:,5),i);
 end

What is wrong with the above statement?

Answer 1

[Image Analyst]

I get "Subscripted assignment dimension mismatch." polyfit() gives you back coefficients, and the number of those is different depending on what order you choose: 2 for linear, 3 for quadratic, 6 for 5th order. So how are you going to stick those all (a varying number) into a column of p? What is the size of P? Try it like this:

N = 5;
% Create some sample data.
time = rand(20,3);
values = rand(20,5);
p = zeros(N, 6); % Preallocate the largest you think you'll need
for k=1:N
  [p(k,1:k+1), S, mu] = polyfit(time(:,3),values(:,5),k);
end

If it still complains, you'll have to use S and mu when you go to use polyval().

[T]

What determines the number 6?

What if I want arbitrary N?

Also what if I want to iterate p?

for k=1:N
  [p(k,1:k+1), S, mu] = polyfit(time(:,3),values(:,5),k);
 f(k,1:k+1) = polyval(p(k,1:k+1),time(:,3));
 plot(time(:,3),f(k,1:k+1))
end

gives me the following error:

??? Subscripted assignment dimension mismatch.

[Image Analyst]

Use

p = zeros(N, N+1); % Preallocate the largest you think you'll need

It's 6 because a 5th order polynomial will need 6 coefficients - one for each "x" term and one for the constant offset.

[T]

Right. But if I want to interpolate the data and plot polyval, how come what I wrote above gives me the error? That is,

for k=1:N
  [p(k,1:k+1), S, mu] = polyfit(time(:,3),values(:,5),k);
      plot = polyval(polyfit(time(:,3),values(:,5),k),time(:,3));
end

If I remove (k,1:k+1) from f(k,1:k+1) and plot using 'hold on' it works but I still get the 'Polynomial is badly conditioned' message.

[T]

Was I not clear?

Answer 2

[Jan]

Nothing is wrong. "Badly conditioned" means that the solution of the system of linear equations critically depends on rounding errors due to the limited precision.

A valid solution (not a workaround only) is the scaling: Transform the polynomial such that the X-values have a mean of 0 and a standard deviation (or range) of 1. Fortunately polyfit does this for you, when you obtain the 3rd output also, see doc polyfit.

Example:

x  = 1000:1004;
y  = rand(1, 5);
p1 = polyfit(x, y, 3);           % Warning appears: Badly conditioned
p2 = polyfit(x - 1000, y, 3);    % No warning
[p3, S, mu] = polyfit(x, y, 3);  % No warning

But of course P2 and p3 differ from p1 now. You can easily apply the transformation to the inputs manually, or automatically in polyval.